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Mirrors > Home > ILE Home > Th. List > m1expeven | GIF version |
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
Ref | Expression |
---|---|
m1expeven | ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9083 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | 2timesd 8986 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) = (𝑁 + 𝑁)) |
3 | 2 | oveq2d 5798 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁))) |
4 | neg1cn 8849 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | neg1ap0 8853 | . . . 4 ⊢ -1 # 0 | |
6 | expaddzap 10368 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 # 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) | |
7 | 4, 5, 6 | mpanl12 433 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
8 | 7 | anidms 395 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
9 | m1expcl2 10346 | . . 3 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | |
10 | neg1rr 8850 | . . . . . 6 ⊢ -1 ∈ ℝ | |
11 | reexpclzap 10344 | . . . . . 6 ⊢ ((-1 ∈ ℝ ∧ -1 # 0 ∧ 𝑁 ∈ ℤ) → (-1↑𝑁) ∈ ℝ) | |
12 | 10, 5, 11 | mp3an12 1306 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℝ) |
13 | elprg 3552 | . . . . 5 ⊢ ((-1↑𝑁) ∈ ℝ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) |
15 | oveq12 5791 | . . . . . . 7 ⊢ (((-1↑𝑁) = -1 ∧ (-1↑𝑁) = -1) → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) | |
16 | 15 | anidms 395 | . . . . . 6 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) |
17 | neg1mulneg1e1 8956 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
18 | 16, 17 | eqtrdi 2189 | . . . . 5 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
19 | oveq12 5791 | . . . . . . 7 ⊢ (((-1↑𝑁) = 1 ∧ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) | |
20 | 19 | anidms 395 | . . . . . 6 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) |
21 | 1t1e1 8896 | . . . . . 6 ⊢ (1 · 1) = 1 | |
22 | 20, 21 | eqtrdi 2189 | . . . . 5 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
23 | 18, 22 | jaoi 706 | . . . 4 ⊢ (((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
24 | 14, 23 | syl6bi 162 | . . 3 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} → ((-1↑𝑁) · (-1↑𝑁)) = 1)) |
25 | 9, 24 | mpd 13 | . 2 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
26 | 3, 8, 25 | 3eqtrd 2177 | 1 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 {cpr 3533 class class class wbr 3937 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 -cneg 7958 # cap 8367 2c2 8795 ℤcz 9078 ↑cexp 10323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 df-exp 10324 |
This theorem is referenced by: m1expe 11632 m1expo 11633 m1exp1 11634 |
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